By the Stoyanov and Prodanov Theorem [1]}, any abelian minimal group is pre-compact. Hence Polish abelian minimal groups are compact.
In this paper, we are interested in locally compact Polish groups, for sake of completeness, we provide a short argument in the case \(G\)  is locally compact abelian.
Let \(\beta \colon G\rightarrow \beta G\)  be the Bohr compactification of the locally compact abelian group \(G\) , see [2]}.
Since minimal groups are homeomorphic to their images, we get that \(G\)  is homeomorphic to \(\beta (G)\) , and since \(G\)  is locally compact, \(\beta (G)\)  is locally compact and hence closed, implying that \(G\)  is compact.
